Mark scheme Pure Mathematics Year 1 (AS) Unit Test 8: Exponentials and Logarithms

Transcription

1 a Substitutes (, 00) into the equation. Substitutes (5, 50) into the equation. Makes an attempt to solve the expressions by division. For 3 example, b (or equivalent) seen ab 6th 5 50 ab Solves for b. b = 0.5 or b A.b Solves for a. a = 600 A.b (5) b Divides by 600 and takes logs of both sides. x k log log 600 Mft.b 5th Understand and use the three laws of logarithms. Uses the third law of logarithms to write or log x xlog anywhere in solution. x log xlog B. Uses the law(s) of logarithms to write anywhere in solution. log log B. Uses above to obtain 600 log k x * log A*. () (9 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

2 Uses appropriate law of logarithms to write log x x Inverse log (or to the) both sides. x x Derives a 3 term quadratic equation. M 3.a 5th x 7x5 0 Solve simple logarithmic equations using the laws of logs. Correctly factorises x x or uses appropriate technique to solve their quadratic. Solves to find 3 x Understands that x 5stating that this solution would require taking the log of a negative number, which is not possible. A.b B 3. (6 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

3 3a Figure Graph has correct shape and does not touch x-axis. The point (0, ) is given or labelled. M 3.a 3rd A 3.a Sketch the graph of y = a x (for a > ) 3bi ii Translation unit right (or positive x direction) or by 0 0 Translation 5 units up (or positive y direction) or by 5 () B.a 5th B.a () Transform the graphs of functions using translations and stretches. ( marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

4 x x Correctly factorises. (or for example, y y States that 8 x, ) 5th 8 x 6 (or y =, y = 6). A.b Solve equations using logarithms. Makes an attempt to solve either equation (e.g. uses laws of indices. For example, 3 8 or 83 or or 83 6 (or correctly takes logs of both sides). Solves to find Solves to find x o.e. or awrt x o.e. or awrt.33 3 M.a A.b A.b (5) (5 marks) nd M mark can be implied by either x or 3 x 3 Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

5 5a Figure Attempt to find intersection with x-axis. For example, log xa 0 9 Solving x a log9 0 to find x = a +, so coordinates of x-intercept are ( a +, 0) oe Substituting x = 0 to derive y log 9 x a, so coordinates of y-intercept are 9 0,log x a Asymptote shown at x = a stated or shown on graph. Increasing log graph shown with asymptotic behaviour and single x-intercept. Fully correct graph with correct asymptote, all points labelled and correct shape. th A.b B 3.a B 3.a M 3.a A.a Sketch the graph y = log(x). 5a 5b log x a log x a seen. M. 5th 9 9 The graph of y x a log 9 is a stretch, parallel to the y- axis, scale factor, of the graph of y x a log 9. (6) A.a () Understand and use the three laws of logarithms. Award all 5 points for a fully correct graph with asymptote and all points labelled, even if all working is not present (8 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

6 6a Makes an attempt to subsitute 7 into the equation, for example, 0. 7 P00e seen. 6 or 60 only (do not accept non-integeric final answer). A 3. th Understand the properties of functions of the form a x. 6b It is the initial bacteria population. B.a th () () Understand the properties of functions of the form a x. 6c 0.t States that 00e or that Solves to find ln 0000 t 0. 0.t e 0000 M 3. 6th (hours) cao (do not accept e.g..0). A 3.5 (3) (6 marks) Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

7 7a 7b Uses the equation of a straight line in the form logv mt c or log V k m( t t0) o.e. Makes correct substitution. logv t log0000 o.e. 0 Either correctly rearranges their equation by exponentiation tlog 0000 For example, V 0 or takes the log of both sides t of the equation V ab. For example, logv log ( ab t ). Completes rearrangement so that both equations are in directly comparable form V t and V ab or logv t log0000 and log V log a t log b. 0 t 6th A.b () 6th States that a = A.b States that b 0 A.b 7c a is the initial value of the car o.e. B.a 6th b is the annual proportional decrease in the value of the car o.e. (allow if explained in figures using their b. For example, (since b is 0.87) the car loses 3% of its value each year.) () B.a () Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.

8 7d Substitutes 7 into their formula from part b. Correct answer is 5 57, accept awrt Bft 3. th () Understand the properties of functions of the form a x. 7e t Uses 0000 ab with their values of a and b or writes log0000 log t (could be inequality). Solves to find t = 0 years. Aft.b M 3. 5th () Solve equations using logarithms. 7f Acceptable answers include. The model is not necessarily valid for larger values of t. Value of the car is not necessarily just related to age. Mileage (or other factors) will affect the value of the car. B 3.5b 6th () ( marks) 7b nd M mark can be implied by correct values of a and b. 7c Accept answers that are the equivalent mathematically. For example, for b. the value of the car in 87% of the value the previous year. Education Ltd 07. Copying permitted for purchasing institution only. This material is not copyright free.